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A Glimpse Into Discrete Differential Geometry

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A Glimpse Into Discrete Differential Geometry COMMUNICATION A Glimpse into Discrete Differential Geometry Keenan Crane and Max Wardetzky Communicated by Joel Hass EDITOR’S NOTE. The organizers of the two-day AMS Short Course on Discrete Differential Geometry have kindly agreed to provide this introduction to the subject. The AMS Short Course runs in conjunction with the 2018 Joint Mathematics Meetings. The emerging field of discrete differential geometry (DDG) studies discrete analogues of smooth geometric objects, providing an essential link between analytical descrip- tions and computation. In recent years it has unearthed a rich variety of new perspectives on applied problems in computational anatomy/biology, computational mechan- Figure 1. Discrete differential geometry reimagines ics, industrial design, computational architecture, and classical ideas from differential geometry without digital geometry processing at large. The basic philoso- reference to differential calculus. For instance, phy of discrete differential geometry is that a discrete surfaces parameterized by principal curvature lines object like a polyhedron is not merely an approximation are replaced by meshes made of circular of a smooth one, but rather a differential geometric object quadrilaterals (top left), the maximum principle in its own right. In contrast to traditional numerical anal- obeyed by harmonic functions is expressed via ysis which focuses on eliminating approximation error conditions on the geometry of a triangulation (top in the limit of refinement (e.g., by taking smaller and right), and complex-analytic functions can be smaller finite differences), DDG places an emphasis on replaced by so-called circle packings that preserve the so-called “mimetic” viewpoint, where key properties tangency relationships (bottom). These discrete of a system are preserved exactly, independent of how surrogates provide a bridge between geometry and large or small the elements of a mesh might be. Just computation, while at the same time preserving as algorithms for simulating mechanical systems might important structural properties and theorems. seek to exactly preserve physical invariants such as total energy or momentum, structure-preserving models of Keenan Crane is assistant professor of computer science at Carnegie Mellon University. His e-mail address is kmcrane@cs discrete geometry seek to exactly preserve global geo- .cmu.edu. metric invariants such as total curvature. More broadly, Max Wardetzky is professor of mathematics at University of Göt- DDG focuses on the discretization of objects that do not tingen. His e-mail address is [email protected] naturally fall under the umbrella of traditional numerical .de. For permission to reprint this article, please contact: analysis. This article provides an overview of some of the [email protected]. themes in DDG. DOI: http://dx.doi.org/10.1090/noti1578 November 2017 Notices of the AMS 1153 COMMUNICATION basic approach of DDG is to find alternative characteriza- tions in the smooth setting that can be applied to discrete geometry in a natural way. With curvature, for instance, we can apply the fundamental theorem of calculus to Equation 1 to acquire a different statement: if 휑 is the angle from the horizontal line to the unit tangent 푇, then 푏 ∫ 휅 푑푠 = 휑(푏) − 휑(푎) mod 2휋. 푎 Put more simply: curvature is the rate at which the tangent Figure 2. A given geometric quantity from the smooth turns. This characterization can be applied naturally to setting, like curvature 휅, may have several reasonable our polygonal curve: along any edge the change in angle definitions in the discrete setting. Discrete is clearly zero. At a vertex it is simply the turning angle differential geometry seeks definitions that exactly 휃푖 ∶= 휑푖,푖+1 − 휑푖−1,푖 between the directions 휑푖−1,푖, 휑푖,푖+1 replicate properties of their smooth counterparts. of the two incident edges, yielding our first notion of discrete curvature: 퐴 (2) 휅푖 ∶= 휃푖 ∈ (−휋, 휋). The Game Are there other characterizations that also lead naturally Our article is organized around a “game” often played in to a discrete formulation? Yes: for instance we can discrete differential geometry in order to come up with a consider the motion of 훾 that most quickly reduces its discrete analogue of a given smooth object or theory: length. In the smooth case it is well known that the (1) Write down several equivalent definitions in the change in length with respect to a smooth variation smooth setting. 휂(푠) ∶ [0, 퐿] → ℝ2 that vanishes at the endpoints of the (2) Apply each smooth definition to an object in the curve is given by integration against curvature: discrete setting. 푑 퐿 (3) Analyze trade-offs among the resulting discrete | length(훾 + 휀휂) = − ∫ ⟨휂(푠), 휅(푠)푁(푠)⟩ 푑푠. definitions, which are invariably inequivalent. 푑휀 휀=0 0 Most often, none of the resulting discrete objects preserve Hence, the velocity that most quickly reduces length is all the properties of the original smooth one—a so- 휅푁. For a polygonal curve, we can simply differentiate 푛−1 called no free lunch scenario. Nonetheless, the properties the sum of the edge lengths 퐿 ∶= ∑푖=1 |훾푖+1 − 훾푖| with that are preserved often prove invaluable for particular respect to any vertex position. At a vertex 푖 we obtain applications and algorithms. Moreover, this activity yields 훾푖 − 훾푖−1 훾푖+1 − 훾푖 (3) 휕훾푖 퐿 = − =∶ 푇푖−1,푖 − 푇푖,푖+1, some beautiful and unexpected consequences—such as |훾푖 − 훾푖−1| |훾푖+1 − 훾푖| a connection between conformal geometry and pure i.e., just a difference of unit tangent vectors 푇푖,푖+1 along combinatorics, or a description of constant-curvature 2 consecutive edges. If 푁푖 ∈ ℝ is the unit angle bisector at surfaces that requires no definition of curvature! To vertex 푖, this difference can also be expressed as highlight some of the challenges and themes commonly 퐵 encountered in DDG, we first consider the simple example (4) 휅푖 푁푖 ∶= 2 sin(휃푖/2)푁푖, of the curvature of a plane curve. Discrete Curvature of Planar Curves How do you define the curvature of a discrete curve? For a smooth arc-length parameterized curve 훾(푠) ∶ [0, 퐿] → ℝ2, curvature 휅 is classically expressed in terms of second 푑 derivatives. In particular, if 훾 has unit tangent 푇 ∶= 푑푠 훾 and unit normal 푁 (obtained by rotating 푇 a quarter turn in the counter-clockwise direction), then 푑2 푑 (1) 휅 ∶= ⟨푁, 푑푠2 훾⟩ = ⟨푁, 푑푠 푇⟩ . Suppose instead we have a polygonal curve with vertices 2 훾1, … , 훾푛 ∈ ℝ , as often used for numerical computation (see Figure 2, right). Here we hit upon the most elemen- tary problem of discrete differential geometry: discrete geometric objects are often not sufficiently differentiable Figure 3. Different characterizations of curvature in (in the classical sense) for standard definitions to apply. the smooth setting naturally lead to different notions For instance, our curvature definition (Equation 1) causes of discrete curvature. (Here we abbreviate 푇푖,푖+1 and trouble, since at vertices our discrete curve is not twice 푇푖−1,푖 by 푢 and 푣, respectively.) differentiable, nor does it have well defined normals. The 1154 Notices of the AMS Volume 64, Number 10 COMMUNICATION providing a discretization of the curvature normal 휅푁. may end up with pointwise or integrated quantities in the A closely-related idea is to consider how the length of discrete case. a curve changes if we displace it by a small constant As one might imagine, there are many other possi- amount in the normal direction. As observed by Steiner, ble starting points for obtaining a discrete analogue of the new length of a smooth curve can be expressed as curvature. Eventually, however, all starting points end 퐿 up leading back to the same definitions, suggesting that (5) length(훾 + 휀푁) = length(훾) − 휀 ∫ 휅(푠) 푑푠. there may be only so many possibilities. For example, if 0 휙 ∶ ℝ2 → ℝ is the signed distance from a smooth closed Since this formula holds for any small piece of the curve 훾, then applying the Laplacian Δ yields the curva- curve, it can be used to obtain a notion of curvature ture of its level curves; in particular, Δ휙|휙=0 yields the at each point. How do we define normal offsets in the curvature of 훾. Likewise, if we apply the Laplacian to the polygonal case? At vertices we again encounter the issue signed distance function for a discrete curve, we recover that we have no notion of normals. One idea is to break the 휅퐴 on one side and 휅퐵 on the other. Yet another approach curve into individual edges which can then be translated is the theory of normal cycles (as discussed by Morvan), by 휀 along their respective normal directions. We can then related to the Steiner formula from Equation 5. Here, close the gaps between edges in a variety of ways: using rather than settle on a single normal 푁푖 at each vertex (A) a circular arc of radius 휀, (B) a straight line, or by we consider all unit vectors between the unit normals (C) extending the edges until they intersect (see Figure 3, of the two incident edges, ultimately leading back to the bottom left). If we then calculate the lengths for these first discrete curvature 휅퐴. The theory of normal cycles new curves, we get applies equally well to both smooth and polygonal curves, 푛−1 again reinforcing the perspective that the fundamental length퐴 = length(훾) − 휀 ∑푖=2 휃푖, 푛−1 behavior of geometry is neither inherently smooth nor length퐵 = length(훾) − 휀 ∑푖=2 2 sin(휃푖/2), 푛−1 discrete, but can be well captured in both settings by length = length(훾) − 휀 ∑ 2 tan(휃푖/2). 퐶 푖=2 picking the appropriate ansatz.
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